Problem 51 - Algebra and Trigonometry

Question

\(49-52=\) A phrase that describes a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) at least 5 units from 7

Step-by-Step Solution

Answer

The inequality is \(|x - 7| \geq 5\).

Step 1: Understanding 'At Least'

The phrase 'at least 5 units from 7' means the distance of any number from 7 must be 5 or more. In mathematical terms, this can be translated into a condition involving absolute values.

Step 2: Expressing in Absolute Values

To express 'at least 5 units from 7' as an inequality, we recognize that the absolute distance between any number \(x\) and 7 is represented by \(|x - 7|\). We want this distance to be greater than or equal to 5.

Step 3: Setting Up the Inequality

The requirement that \(x\) is at least 5 units from 7 can be captured by the inequality \(|x - 7| \geq 5\). This captures both situations where \(x\) is 5 or more units greater than 7, and when \(x\) is 5 or more units less than 7.

Key Concepts

Real Numbers

In mathematics, real numbers are a fundamental concept. They include everything from whole numbers to fractions and decimals, making up the vast set of numbers that we use to describe quantities and measurements in the real world. Real numbers cover:

• integers like -1, 0, 1,
• fractions such as 1/2 and -3/4,
• absolute values of numbers,
• irrational numbers like \(\pi\) and \(\sqrt{2}\).

When we solve problems involving absolute value inequalities, we're often dealing with real numbers. These values are represented on the number line, where every point corresponds to a distinct real number. Understanding real numbers serves as a basis for grasping many mathematical operations and concepts.

Distance in Mathematics

Distance in mathematics is a way to describe how far apart two values or points are. In a more intuitive sense, it's like asking how many steps or units are needed to go from one point to another on a number line or plane.

In the context of absolute value inequalities, distance often refers to how far a number is from zero or another fixed point on the number line. The absolute value of a number \(x\) is its distance from zero, denoted by \(|x|\). For example, both -3 and 3 have an absolute value of 3, because they are each 3 units away from zero.

• The concept of distance helps us form inequalities that define regions on the number line.
• For instance, with the inequality \(|x - 7| \geq 5\), we are looking at numbers that are at least 5 units away from the number 7.

By understanding distance, students can better grasp why certain mathematical relationships and inequalities are expressed the way they are.

Mathematical Inequalities

Mathematical inequalities describe a range of values rather than a single value. They are crucial in expressing situations where quantities are not exactly equal but rather greater than, less than, or equal to other quantities.

• The symbols \(<\) and \(>\) denote "less than" and "greater than" respectively.
• Including the equal sign, as in \(\leq\) or \(\geq\), indicates "less than or equal to" and "greater than or equal to" respectively.

When we talk about absolute value inequalities, we're using these symbols to describe the bounds or distances between numbers. For example, in the inequality \(|x - 7| \geq 5\), both \(x \geq 12\) and \(x \leq 2\) would satisfy the inequality because they define two regions on the number line where the distance from 7 is at least 5 units.
Understanding inequalities helps students describe and analyze sets of real numbers that meet specific criteria and conditions.